Abstract

Petrie polygons, especially as they arise in the study of regular
polytopes and Coxeter groups, have been studied by geometers and group
theorists since the early part of the twentieth century. An open
question is the determination of which polyhedra possess Petrie
polygons that are simple closed curves. The current work explores
combinatorial structures in abstract polytopes, called Petrie schemes,
that generalize the notion of a Petrie polygon. It is established
that all of the regular convex polytopes and honeycombs in Euclidean
spaces, as well as all of the Gr\"unbaum--Dress polyhedra, possess
Petrie schemes that are not self-intersecting and thus have Petrie
polygons that are simple closed curves. Partial results are obtained
for several other classes of less symmetric polytopes.